Consider a general partition of the data \((Y,Z)\)  into \(K=2\)  arbitrary subsets, as in the previous section, with \(Y^{(1:2)}=(Y^{(1)},Y^{(2)})\)  and \(Z^{(1:2)}=(Z^{(1)},Z^{(2)})\) .
A belief update \(\psi \)  is order-coherent in the sense of [1]} if the posterior for independent data is the same regardless of whether we update belief from the prior, taking all the data in one tranche, or update with \(Y^{(1)},Z^{(1)}\)  and use the resulting posterior as the prior for a belief update with \(Y^{(2)},Z^{(2)}\) . In our setting with prior-dependence in the loss function we have the following definition.
